Calculus Examples

$2 x {(4 - x)}^{2}$

Step 1

Rewrite ${(4 - x)}^{2}$ as $(4 - x) (4 - x)$ .

$\frac{d}{d x} [2 x ((4 - x) (4 - x))]$

Step 2

Expand $(4 - x) (4 - x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [2 x (4 (4 - x) - x (4 - x))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [2 x (4 \cdot 4 + 4 (- x) - x (4 - x))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [2 x (4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x))]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $4$ by $4$ .

$\frac{d}{d x} [2 x (16 + 4 (- x) - x \cdot 4 - x (- x))]$

Step 3.1.2

Multiply $- 1$ by $4$ .

$\frac{d}{d x} [2 x (16 - 4 x - x \cdot 4 - x (- x))]$

Step 3.1.3

Multiply $4$ by $- 1$ .

$\frac{d}{d x} [2 x (16 - 4 x - 4 x - x (- x))]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [2 x (16 - 4 x - 4 x - 1 \cdot - 1 x \cdot x)]$

Step 3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [2 x (16 - 4 x - 4 x - 1 \cdot - 1 (x \cdot x))]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [2 x (16 - 4 x - 4 x - 1 \cdot - 1 x^{2})]$

Step 3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [2 x (16 - 4 x - 4 x + 1 x^{2})]$

Step 3.1.7

Multiply $x^{2}$ by $1$ .

$\frac{d}{d x} [2 x (16 - 4 x - 4 x + x^{2})]$

Step 3.2

Subtract $4 x$ from $- 4 x$ .

$\frac{d}{d x} [2 x (16 - 8 x + x^{2})]$

Step 4

Since $2$ is constant with respect to $x$ , the derivative of $2 x (16 - 8 x + x^{2})$ with respect to $x$ is $2 \frac{d}{d x} [x (16 - 8 x + x^{2})]$ .

$2 \frac{d}{d x} [x (16 - 8 x + x^{2})]$

Step 5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 16 - 8 x + x^{2}$ .

$2 (x \frac{d}{d x} [16 - 8 x + x^{2}] + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $16 - 8 x + x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [x^{2}]$ .

$2 (x (\frac{d}{d x} [16] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [x^{2}]) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.2

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$2 (x (0 + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [x^{2}]) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.3

Add $0$ and $\frac{d}{d x} [- 8 x]$ .

$2 (x (\frac{d}{d x} [- 8 x] + \frac{d}{d x} [x^{2}]) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$2 (x (- 8 \frac{d}{d x} [x] + \frac{d}{d x} [x^{2}]) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 (x (- 8 \cdot 1 + \frac{d}{d x} [x^{2}]) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.6

Multiply $- 8$ by $1$ .

$2 (x (- 8 + \frac{d}{d x} [x^{2}]) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 (x (- 8 + 2 x) + (16 - 8 x + x^{2}) \frac{d}{d x} [x])$

Step 6.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 (x (- 8 + 2 x) + (16 - 8 x + x^{2}) \cdot 1)$

Step 6.9

Multiply $16 - 8 x + x^{2}$ by $1$ .

$2 (x (- 8 + 2 x) + 16 - 8 x + x^{2})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$2 (x \cdot - 8 + x (2 x) + 16 - 8 x + x^{2})$

Step 7.2

Apply the distributive property.

$2 (x \cdot - 8) + 2 (x (2 x)) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3

Combine terms.

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Step 7.3.1

Move $- 8$ to the left of $x$ .

$2 (- 8 \cdot x) + 2 (x (2 x)) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.2

Multiply $- 8$ by $2$ .

$- 16 x + 2 (x (2 x)) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.3

Raise $x$ to the power of $1$ .

$- 16 x + 2 (2 (x^{1} x)) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.4

Raise $x$ to the power of $1$ .

$- 16 x + 2 (2 (x^{1} x^{1})) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 16 x + 2 (2 x^{1 + 1}) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.6

Add $1$ and $1$ .

$- 16 x + 2 (2 x^{2}) + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.7

Multiply $2$ by $2$ .

$- 16 x + 4 x^{2} + 2 \cdot 16 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.8

Multiply $2$ by $16$ .

$- 16 x + 4 x^{2} + 32 + 2 (- 8 x) + 2 x^{2}$

Step 7.3.9

Multiply $- 8$ by $2$ .

$- 16 x + 4 x^{2} + 32 - 16 x + 2 x^{2}$

Step 7.3.10

Subtract $16 x$ from $- 16 x$ .

$- 32 x + 4 x^{2} + 32 + 2 x^{2}$

Step 7.3.11

Add $4 x^{2}$ and $2 x^{2}$ .

$- 32 x + 6 x^{2} + 32$

Step 7.4

Reorder terms.

$6 x^{2} - 32 x + 32$